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Weyl algebra
In , especially in the field of , a polynomial ring or polynomial algebra is a (which is also a ) formed from the of s in one or more s (traditionally also called ) with coefficients in another , often a . Polynomial rings occur in many parts of mathematics, and the study of their properties was among the main motivations for the development of and . Polynomial rings and their are fundamental in . Many classes of rings, such as s, s, s, , s, s, are generalizations of polynomial rings. A closely related notion is that of the on a , and, more generally, on an . Differential and skew-polynomial rings Other generalizations of polynomials are differential and skew-polynomial rings. A differential polynomial ring is a ring of s formed from a ring R'' and a ''δ of R'' into ''R. This derivation operates on R'', and will be denoted ''X, when viewed as an operator. The elements of R'' also operate on ''R by multiplication. The is denoted as the usual multiplication. It follows that the relation δ''(''ab) = aδ(b'') + ''δ(a'')''b may be rewritten as : X\cdot a = a\cdot X +\delta(a). This relation may be extended to define a skew multiplication between two polynomials in X'' with coefficients in ''R, which make them a non-commutative ring. The standard example, called a , takes R'' to be a (usual) polynomial ring ''k[Y''], and ''δ to be the standard polynomial derivative \tfrac{\partial}{\partial Y} . Taking a'' =''Y in the above relation, one gets the , X''·''Y − Y''·''X = 1. Extending this relation by associativity and distributivity allows explicitly constructing the . . The skew-polynomial ring is defined similarly for a ring R'' and a ring endomorphism ''f of R'', by extending the multiplication from the relation ''X·''r'' = f''(''r)·''X'' to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism F'' from the monoid '''N' of the positive integers into the endomorphism ring of R'', the formula ''Xn''·''r = F''(''n)(r'')·''Xn'' allows constructing a skew-polynomial ring. Skew polynomial rings are closely related to algebras. Ore extension Weyl algebra In , the '''Weyl algebra' is the of s with coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_{m-1}(X) \partial_X^{m-1} + \cdots + f_1(X) \partial_X + f_0(X). More precisely, let F'' be the underlying , and let ''F[X''] be the in one variable, ''X, with coefficients in F''. Then each ''fi lies in F''[''X]. ∂X is the with respect to X''. The algebra is generated by ''X and ∂X . The Weyl algebra is an example of a that is not a over a . It is also a noncommutative example of a , and an example of an . The Weyl algebra is isomorphic to the of the on two generators, X'' and ''Y, by the generated by the element : YX - XY - 1~. The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in n'' variables. It is generated by ''Xi and ∂Xi, . Weyl algebras are named after , who introduced them to study the in . It is a of the of the , the of the , by setting the central element of the Heisenberg algebra (namely [''X,Y'']) equal to the unit of the universal enveloping algebra (called 1 above). The Weyl algebra is also referred to as the '''symplectic Clifford algebra'. Weyl algebras represent the same structure for symplectic s that s represent for non-degenerate symmetric bilinear forms. Symmetric algebra s. Indeed, if then the Clifford algebra is just the exterior algebra ⋀(V'').}} For nonzero ''Q there exists a canonical linear isomorphism between ⋀(V'') and whenever the ground field ''K does not have characteristic two. That is, they are as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). since it makes use of the extra information provided by Q''.}} The Clifford algebra is a , the is the exterior algebra. ) of the exterior algebra, in the same way that the is a quantization of the .}} Weyl algebras and Clifford algebras admit a further structure of a , and can be unified as even and odd terms of a , as discussed in . * , a of the symmetric algebra by a * , a of the exterior algebra by a Generalizations of Pauli matrices The Pauli matrices \sigma _1 and \sigma _3 satisfy the following: : \sigma _1 ^2 = \sigma _3 ^2 = I, \; \sigma _1 \sigma _3 = - \sigma _3 \sigma _1 = e^{\pi i} \sigma _3 \sigma_1. The so-called is : W = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}. Like the Pauli matrices, W'' is both and . \sigma _1, \; \sigma _3 and ''W satisfy the relation : \; \sigma _1 = W \sigma _3 W^* . The goal now is to extend the above to higher dimensions, d'', a problem solved by (1882). Construction: The clock and shift matrices Fix the dimension as before. Let exp(2''πi''/''d'')}}, a root of unity. Since 1}} and , the sum of all roots annuls: : 1 + \omega + \cdots + \omega ^{d-1} = 0 . Integer indices may then be cyclically identified mod . Now define, with Sylvester, the '''shift matrix : \Sigma _1 = \begin{bmatrix} 0 & 0 & 0 & \cdots &0 & 1\\ 1 & 0 & 0 & \cdots & 0 & 0\\ 0 & 1 & 0 & \cdots & 0 & 0\\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots &\vdots &\vdots \\ 0 & 0 &0 & \cdots & 1 & 0\\ \end{bmatrix} and the clock matrix, : \Sigma _3 = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0\\ 0 & \omega & 0 & \cdots & 0\\ 0 & 0 &\omega ^2 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & \omega ^{d-1} \end{bmatrix}. These matrices generalize σ''1 and ''σ''3, respectively. Note that the unitarity and tracelessness of the two Pauli matrices is preserved, but not Hermiticity in dimensions higher than two. Since Pauli matrices describe , Sylvester dubbed the higher-dimensional analogs "nonions", "sedenions", etc. These two matrices are also the cornerstone of '''quantum mechanical dynamics in finite-dimensional vector spaces' as formulated by , and find routine applications in numerous areas of mathematical physics. The clock matrix amounts to the exponential of position in a "clock" of d'' hours, and the shift matrix is just the translation operator in that cyclic vector space, so the exponential of the momentum. They are (finite-dimensional) representations of the corresponding elements of the on a ''d-dimensional Hilbert space. The following relations echo and generalize those of the Pauli matrices: : \Sigma _ 1 ^d = \Sigma _ 3 ^d = I and the braiding relation, : \; \Sigma_3 \Sigma _1 = \omega \Sigma_1 \Sigma _3 = e^{2 \pi i / d} \Sigma_1 \Sigma _3 , the , and can be rewritten as : \; \Sigma_3 \Sigma _1 \Sigma _3^{d-1} \Sigma_1 ^{d-1} = \omega ~. On the other hand, to generalize the Walsh–Hadamard matrix W'', note : W = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & \omega ^{2 -1} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & \omega ^{d -1} \end{bmatrix}. Define, again with Sylvester, the following analog matrix, still denoted by ''W in a slight abuse of notation, : W = \frac{1}{\sqrt{d}} \begin{bmatrix} 1 & 1 & 1 & \cdots & 1\\ 1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)^2}\\ 1 & \omega^{d-2} & \omega^{2(d-2)} & \cdots & \omega^{(d-1)(d-2)}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 &\omega &\omega ^2 & \cdots & \omega^{d-1} \end{bmatrix}~. It is evident that W'' is no longer Hermitian, but is still unitary. Direct calculation yields : \; \Sigma_1 = W \Sigma_3 W^* ~, which is the desired analog result. Thus, , a , arrays the eigenvectors of , which has the same eigenvalues as . When ''d = 2''k'', W'' * is precisely the matrix of the , converting position coordinates to momentum coordinates and vice versa. The complete family of ''d''2 unitary (but non-Hermitian) independent matrices provides Sylvester's well-known trace-orthogonal basis for \mathfrak{gl} (''d,ℂ), known as "nonions" \mathfrak{gl} (3,ℂ), "sedenions" \mathfrak{gl} (4,ℂ), etc... This basis can be systematically connected to the above Hermitian basis. (For instance, the powers of , the , map to linear combinations of the s.) It can further be used to identify \mathfrak{gl} (d'',ℂ) , as , with the algebra of . Ore extension Suppose that R'' is a (not necessarily commutative) , \sigma \colon R \to R is a ring , and \delta\colon R\to R is a ' σ-derivation of R'', which means that \delta is a homomorphism of s satisfying : \delta(r_1 r_2)=\sigma(r_1)\delta(r_2)+\delta(r_1)r_2 . Then the '''Ore extension' Rx;\sigma,\delta , also called a skew polynomial ring, is the obtained by giving the Rx a new multiplication, subject to the identity : x r=\sigma®x + \delta® . If δ'' = 0 (i.e., is the zero map) then the Ore extension is denoted ''R[x''; ''σ]. If σ'' = 1 (i.e., the identity map) then the Ore extension is denoted ''R[x'',δ''] and is called a differential polynomial ring. The s are Ore extensions, with R'' any commutative , ''σ the identity ring endomorphism, and δ the polynomial derivative. s are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of . See also * * References Category:Advanced mathematics